The shortest distance of the point $(a, b, c)$ from the $x$-axis is

When an object of height $12 \ cm$ is placed at a distance from a convex lens, an image of height $18 \ cm$ is formed on a screen. Without changing the positions of the object and the screen, if the lens is moved towards the screen, another clear image is formed on the screen. The height of this image is (in $cm$)

An insect crawls up a hemispherical surface very slowly (see figure). The coefficient of friction between the insect and the surface is $\mu = 1/3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical,the maximum possible value of $\alpha$ is given by

Which of the following is an incorrect pair?

If the volume of a sphere increases at the rate of $2 \pi \text{ cm}^3/\text{s}$, then the rate of increase of its radius (in $\text{cm/s}$), when the volume is $288 \pi \text{ cm}^3$, is

If the area of the auxiliary circle of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ is twice the area of the ellipse,then the eccentricity of the ellipse is